Problem: 4 12-sided dice are rolled. What is the probability that the number of dice showing a two digit number is equal to the number of dice showing a one digit number? Express your answer as a common fraction. (Assume that the numbers on the 12 sides are the numbers from 1 to 12 expressed in decimal.)
Solution: Since 9 out of the 12 possible results are one digit numbers, each die will show a one digit number with probability of $\frac{3}{4}$ and a two digit number with probability of $\frac{1}{4}$. The probability that two particular dice will show 2 two digit numbers and 2 one digit numbers is thus $\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^2$. There are $\binom{4}{2}=6$ ways to select which two dice will show one digit numbers, so we multiply to get the probability that we want: $6\cdot\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^2=\dfrac{54}{256}=\boxed{\dfrac{27}{128}}$.